Abstract

The study of wild spaces, Peano continua that contain points that are not semi-locally simply connected, has been of integral importance in the study of low-dimensional wild topology. Mainly investigated from the view point of algebraic topology, the fundamental group has been found to be an indespensible tool in describing these spaces, and their local behavior. In particular, using methods developed by Cannon, Conner, Eda, and Kent demonstrated that all maps from Peano continua to planar or one-dimensional Peano continua are determined uniquely up to homotopy class by their induced homomorphisms on fundamental groups, and that fundamental groups are a perfect homotopy invariant for one-dimensional and planar Peano continua \cite{16}. Moreover, in \cite{13}, Eda considered constructing wild spaces, which we call berbered spaces, by adding a null sequence of loops to a dense set of a simply connected, and locally connected Peano conintuum. In doing so, Eda discovered that every homomorphism from the Hawaiian earring space to such a berbered space is induced by a continuous map up to conjugation by a path. Eda then demonstrated in \cite{15} that these homomorphisms were enough to reconstruct the original Peano continuum. The results in this paper extend Eda's findings, demonstrating that every homomorphism from a one-dimensional Peano continuum to a berbered planar Peano continuum must be induced by a continuous map up to conjugation by a path. These findings contribute to the broader understanding of the algebraic and topological structure of Peano continua and their classification up to homotopy.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

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